Deriving a model that expresses density-dependent limitations to growth.

1. Deriving a model that expresses density-dependent limitations to growth.

2. This is unchecked growth: where r is a constant In “checked” population growth, the factor multiplied with N can’t be a constant:

3. K s rmax N

4. The Logistic Growth Model 2 parameters: rmax: the intrinsic (or maximal) per capita rate of growth K: the carrying capacity

5. Birth rates can decline with population size. Death rates can increase with population size. d or b bmax dmin N K

6. The Logistic Growth Model Pierre-François Verhulst Belgian mathematician (1804 – 1849)

7. Plug & Play • Excel Worksheets: • Logistic Growth

8. N K rmax t K K Three graphical representations of the Logistic Growth Model dN dt dN N dt N N absolute population growth rate vs. N relative population growth rate vs. N “Time series” or N vs. time

9. K dN dt • Identify on this graph all population sizes where the population growth rate is zero. • Identify the range of population sizes over which population size increases. • Identify the range of population sizes over which population size decreases. N

10. Growth rates are positive here. Population change in this direction: Growth rates are negative here. Population change in this direction: dN dt N K Zero Growth

11. Definition of an “equilibrium state”: The equilibium state of a population is the population size (N*) where the population neither increases nor decreases.

12. Calculate the equilibria of the logistic growth model:

13. The Logistic Growth Model has two equilibria: N* = 0, also called the “trivial equilibrium”. N* = K is not trivial. A population of size K will neither decline of increase.

14. Definition of a “stable equilibrium state”: An equilibrium N* is said to be locally stable if population sizes that are slightly smaller or slightly bigger move back towards the equilibium.

15. Definition of a “stable equilibrium state”: An equilibrium N* is said to be globally stable if population sizes that have any other size than N* move towards the equilibium.

16. (equilibria) dN dt N K

17. This equilibrium is unstable! This equilibrium is globally stable! Growth rates are positive here. Population change in this direction: Growth rates are negative here. Population change in this direction: (equilibria) dN dt N K

18. The Logistic Growth Model has two equilibria: N* = 0, the “trivial equilibrium” is unstable! N* = K is globally stable! That means populations following the logistic growth model will always approach the carrying capacity K.

19. Summary: • The logistic model is the simplest model that expresses limits to population growth. • It assumes that the per capita population growth rate (dN/(Ndt)) declines linearly with population size N. • The logistic model has two parameters: • rmax is called the intrinsic rate of growth. It expresses the maximal growth rate a population can achieve when at low density. • K is called the carrying capacity. It expresses the stable population size where birth rates = death rates. • The logistic model has one non-trivial, globally stable equilibrium in K.